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Cent. Eur. J. Math. • 1-5 Author version
Central European Journal of Mathematics
Title of the article Article category
First Author1∗ , Second Author2† 1 Department of Mathematics, University Full Name, Street, Postal code, City, Country 2 Institute of Mathematics, Research Institution, Street, Postal code, City, Country
Abstract:
An abstract should accompany every article. It should be a brief summary of significant results of the paper. An abstract should give concise information about the content of the core idea of your paper. It should be informative and not only present the general scope of the paper but also indicate the main results and conclusions. The abstract should not exceed 200 words. It should not contain literature citations, allusions to the tables, tables, figures or illustrations. All nonstandard symbols and abbreviations should be defined. In combination with the title and keywords, an abstract is an indicator of the content of the paper.
MSC:
XXXXX, YYYYY
Keywords: Keyword 1 • Keyword 2 • Keyword 3 c Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1.
Introduction
Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text.
2.
Sample section
Sample theorem with citation: ∗ †
E-mail: [email protected] E-mail: [email protected]
1
Title of the article
Theorem 2.1 (Pythagoras [7]). Let a, b, c denote the sides of a triangle. If the angle between a, b equals 90◦ then |a|2 + |b|2 = |c|2 . Moreover, regardless of the assumption on the angle, it holds that |a| + |b| ≥ |c|.
See References for the bibliography style in CEJM. Below is a proposition with a proof.
Proposition 2.1. The only possible real solutions of the quadratic equation ax2 + bx + c = 0 are √ −b − ∆ x1 = 2a
√ −b + ∆ x2 = , 2a
and
where ∆ = b2 − 4ac.
Proof.
Observe that (x − x1 ) (x − x2 ) = −
∆ bx b2 + x2 + + 2 4a a 4 a2
(1)
Taking into account that ∆ = b2 − 4ac, equation (1) leads to
(x − x1 ) (x − x2 ) =
c bx + x2 + a a
(2)
Finally, (2) shows that a (x − x1 ) (x − x2 ) equals the original equation. A polynomial of degree two cannot have more than two roots, therefore x1 , x2 are the only possible solutions of our equation.
Corollary 2.1. If b2 < 4ac then the quadratic equation ax2 + bx + c = 0 has no real solutions.
2.1.
Subsections and sample formulas
This is a subsection. Typical aligned list of formulas with labels, displayed by using the alignenvironment.
(b + a) (2 b − 3 a) = 2 b2 − a b − 3 a2 123 + 321 = 444
The same list without labels.
(b + a) (2 b − 3 a) = 2 b2 − a b − 3 a2 123 + 321 = 444
2
(3) (4)
F. Author, S. Author
Now the same list with custom tags. (b + a) (2 b − 3 a) = 2 b2 − a b − 3 a2
(∗∗)
123 + 321 = 444
(‡)
Another example of aligned formulas 1+1=2
1+2=3
1+3=4
10 + 1 = 11
10 + 2 = 12
10 + 3 = 13
100 + 1 = 101
100 + 2 = 102
100 + 3 = 103
with some text inserted between,
by using the \intertextcommand.
2.1.1.
Polynomials of degree three
This is a “subsubsection”. Below is a complicated formula, which must be divided into several lines using, for instance, the align* environment: √ 3i 1 − x1 = − 2 2 + 9
3
−3 2
!1 √ 3 12 a4 − 12 a3 + 188 a2 − 432 a + 247 2 a3 + 72 a − 81 − 2 54 √ 3i − 12 a2 − 3 2 a 13 − 3 , √ 3 12 a4 −12 a3 +188 a2 −432 a+247 a−81 − 2 a +72 2 54
√ 3i 1 x2 = − 2 2 + 9
3
−3 2
3
3− 2
x3 =
3
3− 2
3
3− 2
√
12 a4 − 12 a3 + 188 a2 − 432 a + 247 2 a3 + 72 a − 81 − 2 54 √ − 23 i − 12 a2 − 3 a 13 − 3 , √ 3 12 a4 −12 a3 +188 a2 −432 a+247 a−81 − 2 a +72 2 54
a2 − 3 9
3
!1 √ 3 12 a4 − 12 a3 + 188 a2 − 432 a + 247 2 a3 + 72 a − 81 − 2 54
+
!1
−3 3 2
√
12 a4 −12 a3 +188 a2 −432 a+247 2
−
2 a3 +72 a−81 54
a 13 − 3
Theorem 2.2. The numbers x1 , x2 , x3 defined above are the only complex solutions of the equation x3 + a x2 + x + 3 (a − 1) = 0.
Proof.
Easy calculations, using Maxima or some other free computer algebra system.
3
Title of the article
Table 1.
Some caption text.
Some title row 1, column 1 row 1, column 2 row 2, column 1 row 2, column 2 row 3, column 1 row 3, column 2 Another title row row row row row
Value 1
1 2 3 4 5
Value 2 Value 3
130 1025 100 2925 2950
30 1 1 1 1
30 15 10 4 2
Other polynomials This is a paragraph with title. One should admit that there is no general formula for the solutions of general polynomial equations and therefore sometimes we must use numerical methods.
Example 2.1. Consider the polynomial v(t) = t3 + t + 1 of degree three. It turns out that its only real root is 3
t=
3− 2
√ 2
31
1 − 2
!1 3
1
− 3
−3 3 2
√
2
31
−
1 2
1 3
There are also other two complex roots1 , which we ignore.
Remark 2.1. The proof of Theorem 2.2 uses some advanced technology, including a computer algebra system. The same applies to Example 2.1. On the other hand, Proposition 2.1 is well known and its proof is rather elementary.
3.
Tables and figures
Table 1 shows how to show some data using the tableenvironment. Figure 1 shows how to use the figureenvironment for displaying graphics, etc.
Acknowledgements The author(s) would like to thank some institutions for support and so on. 1
4
Since the degree of v is 3, we know that there may be at most 3 roots.
F. Author, S. Author
Figure 1.
The graph of y = x sinx in the interval [-50,50], created by wxMaxima 0.7.4.
References
[1] Anninis K., Crabi T.J., Sunday T.J., New methods for parallel computing, In: Lyonvenson S. (Ed.), Proceedings of Computer Science Conference (1-10 Jul. 2007 Haifa Israel), University Press, 2007, 13-179 [2] Author N., Coauthor M., Title of article, J. Some Math., 2007, 56, 243–256 [3] Katish A., The inconsistency of ZFC, preprint available at http://arxiv.org/abs/1234.1234 [4] Kittel S.J., Maria G., Tuke M., Sepran D.J., Smith J., Tadeuszewicz K., et al., New class of measurable functions, J. Real Anal., 1997, 999, 234-255 [5] Nowak P., New axioms for planar geometry, Eastern J. Math., 1999, 1, 324-334, (in Polish) [6] Nowak P., Even better axioms for planar geometry, Eastern J. Math., (in press, in Polish), DOI: 33.1122/321 [7] Pythagoras S., On the squares of sides of certain triangles, J. Ancient Math., 2003, 4, 1–30, (in Greek) [8] Sambrook J., Uncountable abelian groups, In: Sambrook J., Russell D.W. (Eds.), Contributions to Abelian groups, 3rd ed., Nauka, Moscow, 2001 [9] Sambrook J., Russell D.W., Abelian groups, 3rd ed., Nauka, Moscow, 2001
5
Central European Journal of Mathematics
Title of the article Article category
First Author1∗ , Second Author2† 1 Department of Mathematics, University Full Name, Street, Postal code, City, Country 2 Institute of Mathematics, Research Institution, Street, Postal code, City, Country
Abstract:
An abstract should accompany every article. It should be a brief summary of significant results of the paper. An abstract should give concise information about the content of the core idea of your paper. It should be informative and not only present the general scope of the paper but also indicate the main results and conclusions. The abstract should not exceed 200 words. It should not contain literature citations, allusions to the tables, tables, figures or illustrations. All nonstandard symbols and abbreviations should be defined. In combination with the title and keywords, an abstract is an indicator of the content of the paper.
MSC:
XXXXX, YYYYY
Keywords: Keyword 1 • Keyword 2 • Keyword 3 c Versita Warsaw and Springer-Verlag Berlin Heidelberg.
1.
Introduction
Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text. Some introductory text.
2.
Sample section
Sample theorem with citation: ∗ †
E-mail: [email protected] E-mail: [email protected]
1
Title of the article
Theorem 2.1 (Pythagoras [7]). Let a, b, c denote the sides of a triangle. If the angle between a, b equals 90◦ then |a|2 + |b|2 = |c|2 . Moreover, regardless of the assumption on the angle, it holds that |a| + |b| ≥ |c|.
See References for the bibliography style in CEJM. Below is a proposition with a proof.
Proposition 2.1. The only possible real solutions of the quadratic equation ax2 + bx + c = 0 are √ −b − ∆ x1 = 2a
√ −b + ∆ x2 = , 2a
and
where ∆ = b2 − 4ac.
Proof.
Observe that (x − x1 ) (x − x2 ) = −
∆ bx b2 + x2 + + 2 4a a 4 a2
(1)
Taking into account that ∆ = b2 − 4ac, equation (1) leads to
(x − x1 ) (x − x2 ) =
c bx + x2 + a a
(2)
Finally, (2) shows that a (x − x1 ) (x − x2 ) equals the original equation. A polynomial of degree two cannot have more than two roots, therefore x1 , x2 are the only possible solutions of our equation.
Corollary 2.1. If b2 < 4ac then the quadratic equation ax2 + bx + c = 0 has no real solutions.
2.1.
Subsections and sample formulas
This is a subsection. Typical aligned list of formulas with labels, displayed by using the alignenvironment.
(b + a) (2 b − 3 a) = 2 b2 − a b − 3 a2 123 + 321 = 444
The same list without labels.
(b + a) (2 b − 3 a) = 2 b2 − a b − 3 a2 123 + 321 = 444
2
(3) (4)
F. Author, S. Author
Now the same list with custom tags. (b + a) (2 b − 3 a) = 2 b2 − a b − 3 a2
(∗∗)
123 + 321 = 444
(‡)
Another example of aligned formulas 1+1=2
1+2=3
1+3=4
10 + 1 = 11
10 + 2 = 12
10 + 3 = 13
100 + 1 = 101
100 + 2 = 102
100 + 3 = 103
with some text inserted between,
by using the \intertextcommand.
2.1.1.
Polynomials of degree three
This is a “subsubsection”. Below is a complicated formula, which must be divided into several lines using, for instance, the align* environment: √ 3i 1 − x1 = − 2 2 + 9
3
−3 2
!1 √ 3 12 a4 − 12 a3 + 188 a2 − 432 a + 247 2 a3 + 72 a − 81 − 2 54 √ 3i − 12 a2 − 3 2 a 13 − 3 , √ 3 12 a4 −12 a3 +188 a2 −432 a+247 a−81 − 2 a +72 2 54
√ 3i 1 x2 = − 2 2 + 9
3
−3 2
3
3− 2
x3 =
3
3− 2
3
3− 2
√
12 a4 − 12 a3 + 188 a2 − 432 a + 247 2 a3 + 72 a − 81 − 2 54 √ − 23 i − 12 a2 − 3 a 13 − 3 , √ 3 12 a4 −12 a3 +188 a2 −432 a+247 a−81 − 2 a +72 2 54
a2 − 3 9
3
!1 √ 3 12 a4 − 12 a3 + 188 a2 − 432 a + 247 2 a3 + 72 a − 81 − 2 54
+
!1
−3 3 2
√
12 a4 −12 a3 +188 a2 −432 a+247 2
−
2 a3 +72 a−81 54
a 13 − 3
Theorem 2.2. The numbers x1 , x2 , x3 defined above are the only complex solutions of the equation x3 + a x2 + x + 3 (a − 1) = 0.
Proof.
Easy calculations, using Maxima or some other free computer algebra system.
3
Title of the article
Table 1.
Some caption text.
Some title row 1, column 1 row 1, column 2 row 2, column 1 row 2, column 2 row 3, column 1 row 3, column 2 Another title row row row row row
Value 1
1 2 3 4 5
Value 2 Value 3
130 1025 100 2925 2950
30 1 1 1 1
30 15 10 4 2
Other polynomials This is a paragraph with title. One should admit that there is no general formula for the solutions of general polynomial equations and therefore sometimes we must use numerical methods.
Example 2.1. Consider the polynomial v(t) = t3 + t + 1 of degree three. It turns out that its only real root is 3
t=
3− 2
√ 2
31
1 − 2
!1 3
1
− 3
−3 3 2
√
2
31
−
1 2
1 3
There are also other two complex roots1 , which we ignore.
Remark 2.1. The proof of Theorem 2.2 uses some advanced technology, including a computer algebra system. The same applies to Example 2.1. On the other hand, Proposition 2.1 is well known and its proof is rather elementary.
3.
Tables and figures
Table 1 shows how to show some data using the tableenvironment. Figure 1 shows how to use the figureenvironment for displaying graphics, etc.
Acknowledgements The author(s) would like to thank some institutions for support and so on. 1
4
Since the degree of v is 3, we know that there may be at most 3 roots.
F. Author, S. Author
Figure 1.
The graph of y = x sinx in the interval [-50,50], created by wxMaxima 0.7.4.
References
[1] Anninis K., Crabi T.J., Sunday T.J., New methods for parallel computing, In: Lyonvenson S. (Ed.), Proceedings of Computer Science Conference (1-10 Jul. 2007 Haifa Israel), University Press, 2007, 13-179 [2] Author N., Coauthor M., Title of article, J. Some Math., 2007, 56, 243–256 [3] Katish A., The inconsistency of ZFC, preprint available at http://arxiv.org/abs/1234.1234 [4] Kittel S.J., Maria G., Tuke M., Sepran D.J., Smith J., Tadeuszewicz K., et al., New class of measurable functions, J. Real Anal., 1997, 999, 234-255 [5] Nowak P., New axioms for planar geometry, Eastern J. Math., 1999, 1, 324-334, (in Polish) [6] Nowak P., Even better axioms for planar geometry, Eastern J. Math., (in press, in Polish), DOI: 33.1122/321 [7] Pythagoras S., On the squares of sides of certain triangles, J. Ancient Math., 2003, 4, 1–30, (in Greek) [8] Sambrook J., Uncountable abelian groups, In: Sambrook J., Russell D.W. (Eds.), Contributions to Abelian groups, 3rd ed., Nauka, Moscow, 2001 [9] Sambrook J., Russell D.W., Abelian groups, 3rd ed., Nauka, Moscow, 2001
5
